The Brocard points P and P' of the triangle ABC, illustrated in Figure 1, are such that ∠PAB = ∠PBC = ∠PCA = ω and ∠P'BA = ∠P'CB = ∠P'AC = ω. Problem 78.A in the March 1994 edition of the Mathematical Gazette required a proof that the maximum length of PP' is R/2, where R is the circumradius of triangle ABC. The solution given by various solvers in the November 1994 edition of the Gazette is essentially the same as appeared in a paper by H. E. Piggott in the Gazette in 1924. In the present paper various properties of the triangle and its Brocard points are established in terms of the lengths of the sides of the triangle. Some of the results may be new, but the methods used encompass ‘Mathematics Ancient and Modern’ in the guises of homogeneous areal coordinates and computer algebra!

The two articles which follow, which deal with similar themes in quite different ways, arrived on the editorial table within a short time of each other. Since they are complementary, it seems appropriate to present them here under a common heading.

This article sets out to explore some of the connections between two seemingly distinct mathematical objects: trigonometric functions and the integer sequences composed of the Fibonacci and Lucas numbers. It establishes that elements of Fibonacci/Lucas sequences obey identities that are closely related to traditional trigonometric identities. It then exploits this relationship by converting existing trigonometric results into corresponding Fibonacci/Lucas results. Along the way it uses mathematical tools that are not usually associated with either of these objects.

What sentence in this Mathematical Gazette contains seven a’s, four c’s, thirty-two e’s, eight f’s, five g’s, ten h’s, twelve i’s, three l’s, three m’s, fifteen n’s, eight o’s, one q, seven r’s, twenty-seven s’s, twenty-five t’s, four u’s, eight v’s, seven w’s, four y’s & two z’s?

Problems concerning the set

of numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identity

which shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.

It must be remembered that one does not get a leap year at the turn of a century unless the year is divisible by 400.

In the performance of digital communication systems over fading channels, the error analysis is typically modelled using a Gaussian probability distribution. One function central to the analysis is what engineers routinely refer to as the (Gaussian) Q-function and is defined by

1

This is the canonical representation used for the function. In this paper a number of derivations of an important alternative representation for the Q-function known as Craig's formula will be given.

If you enjoy magic squares and their unusual and fascinating numerical properties, you are in good company. Indeed, many a professional mathematician, scientist and amateur has shared the same interest. Technical research work on the properties of magic squares is available aplenty in the literature or in books concerned with the recreational aspects of mathematics. Studies of magic squares naturally lead to some elements of group theory, of lattices, of Latin squares, of partitions, of matrices, of determinants, and so on …

By definition, the general n x n magic square is a square array of n2 numbers {aij; i, j = 1,2,... , n) whose n rows, n columns, and two main diagonals have the same sum s. It is therefore natural to think of a magic square as a matrix and we shall do this in the present note, simply referring to it as a magic matrix.

Call the ends of a ruler A and Z, and (towards A) place on it a mark X. Draw a line BC the length of AX, and a square BCDE. Erect a perpendicular bisector of BC. With centre C and radius CE draw a circle. Now place the ruler (Fig. 1) so that AZ passes through B, with A on the perpendicular bisector of BC, and with X on the circle. Then ∠BAC = π/7.

A few years ago I tried making a model of an object from its photograph. Realising that, of course, one could not take direct measurements from the photograph, I had to try to guess the relative dimensions of the object. I ended up, unfortunately, with a model which did not seem to be in the correct proportions. Would it have been possible, I wondered, to find the precise ratios of the different lengths?

Proofs of the prime number theorem are extremely hard to follow, and leave the impression, at least among amateurs, that the essential property of prime numbers, namely their primeness, plays very little part in the argument.

Simple reasoning, based on the rules of probability, can however give a very fair indication of the way in which the density of primes in the region x varies with x.